...Consider r= cos2theta in polar form.

I understand that if you want the area of one loop you would integrate ( o.5r^2) between -pi/4 to pi/4.

But what is the interpretation of integrating between -pi to pi? ... the area of all four petals

Also do negative r areas cancel out positive r areas like they do in cartesian? they can, when you get an "inner loop" ... graph the cardioid r = 0.5 + cos(t) to see what I'm talking about.